Review Article ABriefReviewonDynamicsofaCrackedRotor · 2.5. Analysis through Nonlinear Dynamics of...

Hindawi Publishing CorporationInternational Journal of Rotating MachineryVolume 2009, Article ID 758108, 6 pagesdoi:10.1155/2009/758108

Review Article

A Brief Review on Dynamics of a Cracked Rotor

Chandan Kumar and Vikas Rastogi

Mechanical Engineering Department, Sant Longowal Institute of Engineering and Technology, Longowal, Punjab 148106, India

Correspondence should be addressed to Vikas Rastogi, [email protected]

Received 20 September 2009; Accepted 15 December 2009

Recommended by Koji Fujimoto

Fatigue crack is an important rotor fault, which can lead to catastrophic failure if undetected properly and in time. Study andInvestigation of dynamics of cracked shafts are continuing since last four decades. Some review papers were also published duringthis period. The aim of this paper is to present a review on recent studies and investigations done on cracked rotor. It is not theintention of the authors to provide all literatures related with the cracked rotor. However, the main emphasis is to provide all themethodologies adopted by various researchers to investigate a cracked rotor. The paper incorporates a candid commentary onvarious methodologies. The paper further deals an extended Lagrangian formulation to investigate dynamics of cracked rotor.

Copyright © 2009 C. Kumar and V. Rastogi. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

1. Introduction

Fatigue cracks have great potential to cause catastrophicfailures in rotating shaft. This fault may interrupt smooth,effective and efficient operation and performance of themachines. This problem has attracted the attention ofresearchers. Through various researches methodology inves-tigation of dynamics of cracked rotor has been done, andwork in this area is still continuing. Review papers arepresented from time to time on this subject. The studyand investigation in the last few decades on cracked rotorsare presented in a review paper by Wauer [1]; Gasch [2]has presented a survey on simple rotor, Dimarogonas [3]has presented a review, and more recently presented bySabnavis et al. [4]. Various methods like Wavelet transform,Finite element method, Nonlinear dynamics, Hilbert-Huangtransform, and so forth, have been applied to study thedynamic behavior of a cracked rotor. Important achieve-ments have been made during these years; the knowledgeof the dynamical behavior of cracked shafts has helped inpredicting presence of crack in a rotor. But there is demandfor more powerful rotor nowadays. Hence more accurateand reliable technique is required to predict the generationand propagation of a crack in a rotating shaft. In this papera candid comment on various approaches is presented. Analternate method based on extended Lagrangian mechanics

to investigate the dynamics of cracked rotor will be discussedin brief.

2. Different Modeling Approaches Applied toInvestigate Cracked Rotor

Study of cracked rotor is continuing since last fourdecades. Different modeling techniques are used by variousresearchers. These approaches can be presented in the nextsubsections.

2.1. HHT (Hilbert-Huang Transform). Guo and Peng [5]used Hilbert-Huang transform for detection and monitoringof crack in transient response of a cracked rotor. Thismethod is particularly useful for identifying very small crackdepths. Hilbert-Huang transform has good potential fornonlinear and nonstationary data analysis, especially fortime-frequency-energy representations. Ramesh Babu et al.[6] have used HHT for detection of a crack. HHT appearsto be a better tool compared to wavelet transform for crackdetection.

2.2. Breathing Mechanism. Many researchers have analyzedthe dynamics of cracked rotor by the approaches based oncrack breathing. The crack gradually opens and closes during

2 International Journal of Rotating Machinery

each revolution, in other words breaths during the revolutionof the shaft; however, it is not yet entirely clear howpartial closure interacts with key variables of the problem.Hence there is a need for a model that accounts for thecrack breathing mechanism and for the interaction betweenexternal loading and dynamic crack behavior concludedby Bachschmid et al. [7]. The crack breathing in rotorsis often simulated using two well-known models by Pateland Darpe [8]. One is the switching/hinge crack model, inwhich the stiffness of the rotor switches from one corre-sponding to the closed crack to the stiffness correspondingto the fully open crack state. The other model is theresponse dependent breathing crack model presented byJun et al. [9].

2.3. Wavelet Transform and Wavelet Finite Element Approach.Wavelets method may be used to detect the damage locationand depth by considering modal frequencies, modal shape,and modal damp, and so forth. Zheng et al. [10] applied thewavelet transform technique for bifurcation and chaos study.Wavelet transform can reveal the local property in both timedomain and frequency domain. They introduced a methodto analyze the existing domains of different types of motionin the parametric space of a nonlinear system. Yang et al.[11] have worked on characterization and detection of crack-induced rotary instability. They developed and presenteda wavelet-based algorithm to characterize periodic, period-doubling, fractural-like, and chaotic motions as results ofthe inherent nonlinearity associated with crack opening andclosing during vibration. The most significant feature of thisalgorithm is that, it is capable of continuously identifyingthe transition state that marks the initiation and propagationof rotor dynamical mechanical stability. Darpe [12] haspresented a method to detect transverse surface crack ina rotating shaft. His method was also based on wavelettransform techniques. Xiang et al. [13] have presented awavelet finite element method for identification of crack ina rotor system and proposed a methodology to analyze thedynamic behavior of a Rayleigh beam.

Traditional FEM has some limitation, such as lowefficiency, insufficient accuracy, slow convergence to correctsolutions, and so forth, in case of complex problem withhigh gradient or strong nonlinearity. In order to overcomethese difficulties, wavelet spaces have been employed asapproximate spaces and then wavelet finite element methods(WFEM) have been derived by Ma et al. [14]. Chen et al. [15,16] applied this technique to a dynamic multi scale liftingcomputation method using Daubechies wavelet. The desir-able advantages of WFEM are multiresolution properties andvarious basis functions for structural analysis. Li et al. [17]proposed a methodology to detect crack location and size,which takes advantage of WFEM in the modal analysis forsingularity problems like a cracked beam. By using WFEMmodel, Xiang et al. [18] gave the experimental identificationresults in cantilever beams with small identification errors.Xiang et al. [19] also proposed a BSWI Euler beam model fordetecting crack in a beam, the simulation and experimentalresults show the high performance of the BSWI Eulerbeam element. The desirable advantages of WFEM are

multiresolution properties and various basis functions forstructural analysis.

2.4. Investigation through Finite Element Approach. Three-dimensional (3D) FEA has attracted the researchers as aninvestigative tool for the study of crack breathing mecha-nism. Due to ease in simulation, many researchers appliedfinite element technique for analysis of crack in a rotatingshaft. Papadopoulos and Dimarogonas [20] have analyzedfor the local crack compliance for a six-degree-of-freedomcrack beam segment. Alternatively the local compliancematrix can be determined through a 3D state finite elementanalysis by several researchers. Sekhar and Prabhu [21]have studied the vibration and stress fluctuation in crackedshafts and investigated a simply supported shaft with atransverse crack for the vibrational characteristics. Finiteelement analysis (FEA) has been carried out for free andforce vibrations. Dirr et al. [22] worked for detection andsimulation of small transverse cracks in rotating shafts. Intheir work, they developed a spatial finite element model andused it for numerical simulation of the dynamic behavior of acracked rotor, in another study. Mohiuddin and Khulief [23]have investigated model characteristics of a cracked rotorusing a conical shaft finite element and deduced frequencyof the rotor in several cracked conditions. Sekhar and BalajiPrasad [24] worked on dynamic analysis of a rotor systemconsidering a slant crack in shaft. FEM analysis of a rotorbearing system for flexural vibration has been considered intheir work and they concluded a general trend of reductionin eigen frequencies of all models with an increase in crackdepth. Sekhar [25] has investigated vibration characteristicsof a cracked rotor with two opens and applied FEM analysison rotor system for flexural vibration considering twotransverse open crack. He concluded that in the case of twocracks of different depths, the larger crack has the moresignificant effect on the eigen frequency of the rotor. Nandi[26] has presented a simple method of reduction for finiteelement model of nonaxisymetric rotors on nonisotropicspring. Darpe [27] investigated the dynamics of a Jeffcottrotor with slant crack describing the flexibility matrix for theslant crack. Bachschmid and Tanzi [28] have looked at thestraight front crack. Recently, Georgantzinos and Anifantis[29] studied the effect of the crack breathing mechanism onthe time-variant flexibility due to the crack in a rotating shaftconsidering quasistatic approximation and using advancednonlinear contact-FEM procedure. This method predicts thepartial contact of crack surfaces, and it is appropriate toevaluate the instantaneous crack flexibilities.

Darpe et al. [30, 31] calculated the breathing by evalu-ating the rectilinear crack tip, the point where the crack isstarting to close, assuming that the closed part of the cracksurface is delimited by a boundary, the “crack closure line”(CCL) represented by a segment, orthogonal to the cracktip, that can be drawn from that point on the crack tip.The same approach has been used also by Wu et al. [32] intime step calculations where breathing was determined byvibrations. Unlike the previous studies, in Chasalevris andPapadopoulos [33] each crack is characterized by its depth,position, and relative angle. The compliance calculation

International Journal of Rotating Machinery 3

method is applied for the first time in cracked shaftswith rotated cracks. The compliance matrix of the crackin bending is calculated using the well-known method ofintegration of the strain energy density function over theopened crack surface.

2.5. Analysis through Nonlinear Dynamics of Cracked Rotor.Muller et al. [34] investigated nonlinear dynamics of acracked rotating shaft using changing stiffness coefficient.Bovsunovskii [35] has done the numerical study of forcedand decaying vibrations of a system simulating a body witha closing crack under the action of various modes of anonlinear restoring force and linear viscous friction. In hisstudy, he determined the natural frequency of transverse andlongitudinal vibration of a cracked beam with an open andclose crack.

Qin et al. [36] have investigated bifurcation in theresponse of cracked Jeffcott rotor. Nonlinear response ofresponse of a cracked rotor due to bifurcation and itsinfluence was illustrated in their study. Feng et al. [37]worked for nonlinear analysis of a cracked rotor withwhirling. Distinct differences have been found in bifurcation,amplitude, orbit, and Poincare map when carrying on thiscomparison. The work may be useful in crack early detectionand diagnosis. Zhu et al. [38] have investigated the dynamicsof a cracked rotor with an active magnetic bearing, throughtheir theoretical analysis they observed that it is impossibleto use traditional method with super harmonic frequencycomponents in the supercritical speed region to detect thecrack. Zhou et al. [39] have analyzed the cracked rotor exper-imentally and investigated nonlinear dynamic characters ofrotor system containing a transverse fatigue crack. Ishida andInoue [40] have used piecewise model function and powerseries function for zetchin of a rotor crack by a harmonicexcitation and nonlinear vibration analysis. Bachschmid etal. [41] evaluated all the nonlinear effects due to a ratherdeep transverse crack that has developed in a full size shaft.Time integration has been used for the nonlinear analysisof a heavy, horizontal axis, and well-damped steam turbinerotor. The results confirmed that neither instabilities norsubharmonic components did appear, and that the overalldeviations from linear behavior were rather small.

2.6. Analysis of Crack Rotor through Several Other Techniques.Besides the above mentioned techniques, there are fewtechniques and methodologies developed by few researchersto analyze the dynamics of cracked rotor. Shulzhenko andOvcharova [42] studied the effect of the break of theelastic axis of rotor with transverse crack on its vibrationalcharacteristics and performed a numerical analysis to knowthe various effects on vibration due to the crack. Bachschmidet al. [43] have proposed a method for identification ofthe position and the depth of a transverse crack in a rotorsystem, by using vibration measurement technique. A modelbased diagnostic approach and a least squares identificationmethod in the frequency domain are used for the cracklocalization along the rotor. Meng and Gasch [44] workedon stability and stability degree of a cracked flexible rotorsupported on journal bearings and concluded that crack

ridge zone depends largely on the speed ratio and the stiffnesschange ratio. Andrieux and Vare [45] worked on a 3Dcracked beam model with unilateral contact, which may beapplied on a rotor. They have given a general procedurefor deriving a lumped cracked section beam model, whichhas been designed with 3D computations and incorporatingmore realistic behavior of the crack, under variable loading.Sekhar [46] has proposed a model-based approach forcracked identification in a rotor system with a suggestion thatthe better expansion techniques or refined modal expansionmethod should be incorporated to get better results and tofinally get this scheme. Sekhar et al. [47] have investigatedthe dynamics of the rotor system with two slant cracks anda transverse crack and found that the effect of transversecrack is highly sensitive to mechanical impedance comparedto slant crack. Gomez-Mancilla et al. [48] have investigatedthe influence of transverse crack in a rotating shaft andnoted that as the crack depth increase, an increase of theorbit completed is observed in the rotor. Huang and Liu[49] have presented a dynamic analysis of rotating beamswith nonuniform cross sections using the dynamic stiffnessmethod. They, however, used Hamilton’s principle for thenumerical analysis and prepared dynamic stiffness matrix fornumerical analysis.

2.7. Recent Developments in Cracked Rotor Analysis. Since thelast two years, few papers have appeared in various archives,where few new methods and tools have been applied for anal-ysis of dynamics of rotor and its correct prediction. Changet al. [50] have studied nonlinear response and dynamicstability of a cracked rotor. They developed nonlineargoverning equations of motion for the cracked rotor systemwith asymmetrical visco elastic supports. Mueller et al. [51]have presented a comparison of different approaches forestimation of a creep crack initiation and compared the timedependent failure assessment diagram (TDFAD), two criteriadiagram (ZCD), and Nikbin-Smith Webster model (NSW-model) approaches for predicting creep crack initiation.It has been found that TDFAD and NSW-model methodare costly but ZCD is a simple method. In another recentliterature, Dong et al. [52] introduced a novel crack detectionmethod based on high precision modal parameter identifica-tion in the BSWI finite element framework, where Empiricalmode decomposition and Laplace wavelet are proposed.From the selected, ten largest correlation coefficients and thecorresponding parameters of natural frequency and dampingare identified. The errors in the WFE model are minimizedby finite element model updating technique, such as outputerror method is for this application. The effectiveness ofthe proposed method is verified by an experiment andcrack parameters are identified. A comprehensive theoretical,numerical, and experimented approach for crack detectionin power plant rotating machinery has been presented byStoisser and Audebert [53]. In this study, they developeda 3D theoretical cracked beam model and presented anumerical simulation, which has been validated throughexperimental results. In a recent study Jun and Gadala [54]investigated the dynamic behavior analysis of cracked rotor,


by considering an additional slope in the crack breathing.In this analysis fracture mechanics concept was used andresponse of cracked rotor was formulated based on transfermatrix method. Recently, Xiang et al. [55] constructed BSWIrotating Rayleigh-Euler and Rayleigh-Timoshenko beamelements to obtain a precision crack detection database,whereas the inverse problem of normalized crack locationand depth is detected using genetic algorithm. Pennacchi andVania [56] have analyzed the shaft vibrations of a 100 MWpower unit. The results obtained with a model-based analysisof the shaft vibrations caused by the propagation of a crackoccurred in the load coupling connected to the gas turbine ofa power unit are discussed.

3. Alternative Proposal for Cracked Rotorthrough Extended Lagrangian Mechanics

Among the various analytical techniques used by variousresearchers, Lagrangian mechanics are one of the area, wherefew works have been reported in literature. It is a well-knownphenomenon that asymmetric rotating component producesnonpotential and dissipative forces, and classical Lagrange’sequation cannot analyze the dynamics of such system withnonholonomic constraints, nonpotential forces, dissipativeforces, gyroscopic forces, and general class of systems withtime fluctuating parameters, as such Lagrangian cannot beworked out due to nonconservative forces involved in thesystem. So some additional information of system interiorand exterior is needed in generating extended Lagrangianequations through bond graphs [57–59], which may beapplicable to the crack rotor system. The symmetries of thesystem also provide useful information to derive the constantof motion of the system as they reduce the complexityof the dynamical systems such as cracked rotor. Noether’stheorem [60] played a significant role in determination ofinvariants of motion. Noether’s theorem allows exploitationof symmetries of the system to arrive at invariant of motion.Extended Noether’s theorem with Umbra’s Hamiltonianprovides the insight of dynamics of the asymmetric rotorsystem.

To enlarge the scope of Lagrangian-Hamiltonian mecha-nics, a new proposal of additional time like variable “umbra-time” was made by Mukherjee [61] and this new concept ofumbra-time leads to a peculiar form of equation, which istermed as umbra-Lagrange’s equation. A brief and candidcommentary on the idea of umbra Lagrangian is given byBrown [62]. This idea was further consolidated by presentingan important issue of invariants of motion for the generalclass of system by extending Noether’s theorem [63]. Thisnotion of Umbra-time is again used to propose a newconcept of umbra-Hamiltonian, which is used along withthe extended Noether’s theorem to provide an insight intothe dynamics of systems with symmetries. One of themost important insights gained from the umbra-Lagrangianformalism is that its underlying variational principle [64]is possible, which is based on their cursive minimization offunctionals. Once such a least action principle is established,many significant results of analytical mechanics may be

extended to a general class of system. In this direction [64]defined all these notions in an extended manifold comprisingof real time, and umbra and real time displacements andvelocities. The umbra-Lagrangian theory has been used suc-cessfully to study invariants of motion for nonconservativemechanical and thermomechanical systems [65]. RecentlyMukherjee et al. [66] applied umbra Lagrangian to studydynamics of an electromechanical system comprising of aninduction motor driving an elastic rotor. In another researchwork, the same authors [67] have provided the dynamics aone-dimensional internally damped rotor through dissipa-tive coupling. Moreover Rastogi and Kumar [68] investigatedthe dynamic behavior of asymmetric rotor by extendingthe Lagrangian-Hamiltonian Mechanics. This is in line tobroaden the scope of umbra-Lagrangian formulation appliedto this rotor dynamic research. In this paper, symmetry of therotor is broken in terms of stiffness only and it may be appliedin other cases also.

4. Conclusions

After an exhaustive survey of literature, the potentialresearchers in this field have various tools and methodologiesto be applied to detect and proper identification of a crack.The methodologies have some specific advantages over theother. Now, there is a choice before the dynamics people toadopt proper techniques. The paper may be concluded asfollows.

(i) Crack breathing mechanism plays an importantrole in analysis of dynamic behavior of a crackedrotor. This breathing phenomena must be modeledaccurately to detect the crack in a rotor.

(ii) Wavelet transform method was used by variousresearchers to predict the generation and propagationof a crack in rotating shaft. This propagation mecha-nism may be modeled so that the early failure of therotor shaft may be detected.

(iii) Finite element method is a better choice and appliedby various researchers to analyze the dynamic behav-ior of a shaft having different kind of cracks, forexample, transverse crack, two cracks, slant crack,and so forth. The crack element must be accuratelydiscretized to depict the real behavior of a crackedrotor.

(iv) Wavelet along with finite element method is also agood choice to have advantages of both techniques.Recently wavelet along with genetic algorithm analy-sis is also finding place in the various Literature.

(v) The proposal of extended lagrangian-Hamiltonianformulation provides the invariants of motion ofthe dynamical system. It provides a great insight ofthe dynamics of a cracked rotor through extendedNoether’s theorem.

International Journal of Rotating Machinery 5

References

[1] J. Wauer, “On the dynamics of cracked rotors: a literaturesurvey,” Applied Mechanics Reviews, vol. 43, pp. 13–17,1990.

[2] R. Gasch, “A survey of the dynamic behavior of a simplerotating shaft with a transverse crack,” Journal of Sound andVibration, vol. 160, no. 2, pp. 313–332, 1983.

[3] A. D. Dimarogonas, “Vibration of cracked structure: a state ofthe art review,” Engineering Fracture Mechanics, vol. 55, no. 5,pp. 831–857, 1996.

[4] G. Sabnavis, R. G. Kirk, M. Kasarda, and D. Quinn, “Crackedshaft detection and diagnostics: a literature review,” Shock andVibration Digest, vol. 36, no. 4, pp. 287–296, 2004.

[5] D. Guo and Z. K. Peng, “Vibration analysis of a crackedrotor using Hilbert-Huang transform,” Mechanical Systemsand Signal Processing, vol. 21, no. 8, pp. 3030–3041, 2007.

[6] T. Ramesh Babu, S. Srikanth, and A. S. Sekhar, “Hilbert-Huang transform for detection and monitoring of crack in atransient rotor,” Mechanical Systems and Signal Processing, vol.22, no. 4, pp. 905–914, 2008.

[7] N. Bachschmid, P. Pennacchi, and E. Tanzi, “Some remarks onbreathing mechanism, on non-linear effects and on slant andhelicoidal cracks,” Mechanical Systems and Signal Processing,vol. 22, no. 4, pp. 879–904, 2008.

[8] T. H. Patel and A. K. Darpe, “Influence of crack breathingmodel on nonlinear dynamics of a cracked rotor,” Journal ofSound and Vibration, vol. 311, no. 3–5, pp. 953–972, 2008.

[9] O. S. Jun, H. J. Eun, Y. Y. Earmme, and C.-W. Lee, “Modellingand vibration analysis of a simple rotor with a breathingcrack,” Journal of Sound and Vibration, vol. 155, no. 2, pp. 273–290, 1992.

[10] L. Zheng, H. Gao, and T. Guo, “Application of wavelet trans-form to bifurcation and chaos study,” Applied Mathematics andMechanics, vol. 19, no. 6, pp. 593–599, 1998.

[11] B. Yang, C. S. Suh, and A. K. Chan, “Characterizationand detection of crack-induced rotary instability,” Journal ofVibration and Acoustics, vol. 124, no. 1, pp. 40–48, 2002.

[12] A. K. Darpe, “A novel way to detect transverse surface crack ina rotating shaft,” Journal of Sound and Vibration, vol. 305, no.1-2, pp. 151–171, 2007.

[13] J. W. Xiang, X. F. Chen, Q. Mo, and Z. J. He, “Identificationof crack in a rotor system based on wavelet finite elementmethod,” Finite Elements in Analysis and Design, vol. 43, no.14, pp. 1068–1081, 2007.

[14] J. X. Ma, J. J. Xue, S. J. Yang, and Z. J. He, “A study of theconstruction and application of a Daubechies wavelet-basedbeam element,” Finite Elements in Analysis and Design, vol. 39,no. 10, pp. 965–975, 2003.

[15] X. F. Chen, S. J. Yang, J. X. Ma, and Z. J. He, “The constructionof wavelet finite element and its applications,” Finite Elementsin Analysis and Design, vol. 40, no. 5-6, pp. 541–554, 2004.

[16] X. F. Chen, Z. J. He, J. W. Xiang, and B. Li, “A dynamicmultiscale lifting computation method using Daubechieswavelet,” Journal of Computational and Applied Mathematics,vol. 188, no. 2, pp. 228–245, 2006.

[17] B. Li, X. F. Chen, J. X. Ma, and Z. J. He, “Detection of cracklocation and size in structures using wavelet finite elementmethods,” Journal of Sound and Vibration, vol. 285, no. 4-5,pp. 767–782, 2005.

[18] J. W. Xiang, X. F. Chen, B. Li, Y. M. He, and Z. J. He,“Identification of a crack in a beam based on the finite elementmethod of a B-spline wavelet on the interval,” Journal of Soundand Vibration, vol. 296, no. 4-5, pp. 1046–1052, 2006.

[19] J. W. Xiang, X. F. Chen, Q. Mo, and Z. J. He, “Identificationof crack in a rotor system based on wavelet finite elementmethod,” Finite Elements in Analysis and Design, vol. 43, no.14, pp. 1068–1081, 2007.

[20] C. A. Papadopoulos and A. D. Dimarogonas, “Coupling ofbending and torsional vibration of a cracked Timoshenkoshaft,” Ingenieur-Archiv, vol. 57, no. 4, pp. 257–266, 1987.

[21] A. S. Sekhar and B. S. Prabhu, “Vibration and stress fluctu-ation in cracked shafts,” Journal of Sound and Vibration, vol.169, no. 5, pp. 655–667, 1994.

[22] B. O. Dirr, K. Popp, and W. Rothkegel, “Detection andsimulation of small trasverse cracks in rotating shafts,” Archiveof Applied Mechanics, vol. 64, no. 3, pp. 206–222, 1994.

[23] M. A. Mohiuddin and Y. A. Khulief, “Modal characteristics ofcracked rotors using a conical shaft finite element,” ComputerMethods in Applied Mechanics and Engineering, vol. 162, no.1–4, pp. 223–247, 1998.

[24] A. S. Sekhar and P. Balaji Prasad, “Dynamic analysis of a rotorsystem considering a slant crack in the shaft,” Journal of Soundand Vibration, vol. 208, no. 3, pp. 457–473, 1997.

[25] A. S. Sekhar, “Vibration characteristics of a cracked rotor withtwo open cracks,” Journal of Sound and Vibration, vol. 223, no.4, pp. 497–512, 1999.

[26] A. Nandi, “Reduction of finite element equations for a rotormodel on non-isotropic spring support in a rotating frame,”Finite Elements in Analysis and Design, vol. 40, no. 9-10, pp.935–952, 2004.

[27] A. K. Darpe, “Dynamics of a Jeffcott rotor with slant crack,”Journal of Sound and Vibration, vol. 303, no. 1-2, pp. 1–28,2007.

[28] N. Bachschmid and E. Tanzi, “Deflections and strains incracked shafts due to rotating loads: a numerical and exper-imental analysis,” International Journal of Rotating Machinery,vol. 10, pp. 283–291, 2004.

[29] S. K. Georgantzinos and N. K. Anifantis, “An insight into thebreathing mechanism of a crack in a rotating shaft,” Journal ofSound and Vibration, vol. 318, no. 1-2, pp. 279–295, 2008.

[30] A. K. Darpe, K. Gupta, and A. Chawla, “Dynamics of a bowedrotor with a transverse surface crack,” Journal of Sound andVibration, vol. 296, no. 4-5, pp. 888–907, 2006.

[31] A. K. Darpe, “Coupled vibrations of a rotor with slant crack,”Journal of Sound and Vibration, vol. 305, no. 1-2, pp. 172–193,2007.

[32] X. Wu, M. I. Friswell, J. T. Sawicki, and G. Y. Baaklini, “Finiteelement analysis of coupled lateral and torsional vibrations ofa rotor with multiple cracks,” in Proceedings of the ASME TurboExpo 2005: Gas Turbine Technology: Focus for the Future, vol. 4,pp. 841–850, Reno, Nev, USA, June 2005.

[33] A. C. Chasalevris and C. A. Papadopoulos, “Identification ofmultiple cracks in beams under bending,” Mechanical Systemsand Signal Processing, vol. 20, no. 7, pp. 1631–1673, 2006.

[34] P. C. Muller, J. Bajkowski, and D. Soffker, “Chaotic motionsand fault detection in a cracked rotor,” Nonlinear Dynamics,vol. 5, no. 2, pp. 233–254, 1994.

[35] A. P. Bovsunovskii, “On determination of the natural fre-quency of transverse and longitudinal vibrations of a crackedbeam,” Strength of Materials, vol. 31, no. 2, pp. 130–137, 1999.

[36] W. Qin, G. Chen, and X. Ren, “Grazing bifurcation in theresponse of cracked Jeffcott rotor,” Nonlinear Dynamics, vol.35, no. 2, pp. 147–157, 2004.

[37] L. X. Feng, X. U. Ping-Youn, S. Lin, and S. Yang, “Non linearanalysis of a cracked rotor with whirling,” Applied Mathematicsand Mechanics, vol. 23, pp. 721–731, 2002.


[38] C. Zhu, D. A. Robb, and D. J. Ewins, “The dynamics of acracked rotor with an active magnetic bearing,” Journal ofSound and Vibration, vol. 265, no. 3, pp. 469–487, 2003.

[39] T. Zhou, Z. Sun, J. Xu, and W. Han, “Experimental analysis ofcracked rotor,” Journal of Dynamic Systems, Measurement andControl, vol. 127, no. 3, pp. 313–320, 2005.

[40] Y. Ishida and T. Inoue, “Detection of a rotor crack using aharmonic excitation and nonlinear vibration analysis,” Journalof Vibration and Acoustics, vol. 128, no. 6, pp. 741–749, 2006.

[41] N. Bachschmid, P. Pennacchi, and E. Tanzi, “Some remarks onbreathing mechanism, on non-linear effects and on slant andhelicoidal cracks,” Mechanical Systems and Signal Processing,vol. 22, no. 4, pp. 879–904, 2008.

[42] N. G. Shulzhenko and G. B. Ovcharova, “ Effect of break of theelastic axis of a rotor with transverse crack on its vibrationalcharacteristics,” Strength of Material, vol. 29, no. 4, pp. 380–385, 1997.

[43] N. Bachschamid, P. Pennacchie, E. Tanzi, and Avana, “Iden-tification of transverse crack position and depth in motorsystem,” Meccanica, vol. 35, pp. 563–582, 2000.

[44] G. Meng and R. Gasch, “Stability and stability degree of acracked flexible rotor supported on journal bearings,” Journalof Vibration and Acoustics, vol. 122, no. 2, pp. 116–125, 2000.

[45] S. Andrieux and C. Vare, “A 3D cracked beam model withunilateral contact. Application to rotors,” European Journal ofMechanics, A/Solids, vol. 21, no. 5, pp. 793–810, 2002.

[46] A. S. Sekhar, “Crack identification in a rotor system: a model-based approach,” Journal of Sound and Vibration, vol. 270, no.4-5, pp. 887–902, 2004.

[47] A. S. Sekhar, A. R. Mohanty, and S. Prabhakar, “Vibrationsof cracked rotor system: transverse crack versus slant crack,”Journal of Sound and Vibration, vol. 279, no. 3–5, pp. 1203–1217, 2005.

[48] J. Gomez-Mancilla, J.-J. Sinou, V. R. Nosov, F. Thouverez, andA. Zambrano, “The influence of crack-imbalance orientationand orbital evolution for an extended cracked Jeffcott rotor,”Comptes Rendus Mecanique, vol. 332, no. 12, pp. 955–962,2004.

[49] K. J. Huang and T. S. Liu, “Dynamic analysis of rotating beamswith nonuniform cross sections using the dynamic stiffnessmethod,” Journal of Vibration and Acoustics, vol. 123, no. 4,pp. 536–539, 2001.

[50] P. C. Chang, D. Liming, and F. Yiming, “Non linear responseand dynamic stability of a cracked rotor,” Communications inNonlinear Science and Numerical Simulation, vol. 12, pp. 1023–1037, 2007.

[51] F. Mueller, A. Scholz, and C. Berger, “Comparison of differentapproaches for estimation of creep crack initiation,” Engineer-ing Failure Analysis, vol. 14, no. 8, pp. 1574–1585, 2007.

[52] H. B. Dong, X. F. Chen, B. Li, K. Y. Qi, and Z. J. He, “Rotorcrack detection based on high-precision modal parameteridentification method and wavelet finite element model,”Mechanical Systems and Signal Processing, vol. 23, no. 3, pp.869–883, 2009.

[53] C. M. Stoisser and S. Audebert, “A comprehensive theoretical,numerical and experimental approach for crack detectionin power plant rotating machinery,” Mechanical Systems andSignal Processing, vol. 22, pp. 818–844, 2008.

[54] O. S. Jun and M. S. Gadala, “Dynamic behavior analysis ofcracked rotor,” Journal of Sound and Vibration, vol. 309, no.1-2, pp. 210–245, 2008.

[55] J. W. Xiang, Y. Zhong, X. F. Chen, and Z. J. He, “Crackdetection in a shaft by combination of wavelet-based elementsand genetic algorithm,” International Journal of Solids andStructures, vol. 45, no. 17, pp. 4782–4795, 2008.

[56] P. Pennacchi and A. Vania, “Diagnostics of a crack in a loadcoupling of a gas turbine using the machine model and theanalysis of the shaft vibrations,” Mechanical Systems and SignalProcessing, vol. 22, no. 5, pp. 1157–1178, 2008.

[57] D. C. Karnopp, R. C. Rosenberg, and D. L. Margolis, SystemDynamics: A Unified Approach, John Wiley & Sons, New York,NY, USA, 1990.

[58] P. C. Breedveld and G. Dauphin-Tanguy, Bond Graphs for Engi-neers, North-Holland, Amsterdam, The Netherlands, 1992.

[59] P. Gawthrop and L. Smith, MetaModeling: Bond Graphs andDynamic Systems, Prentice-Hall, Upper Saddle River, NJ, USA,1996.

[60] E. Noether, “Invariant variationsprobleme,” Ges. Wiss. Wor-thington, vol. 2, pp. 235–257, 1918.

[61] A. Mukherjee, “Junction structures of bondgraph theory fromanalytical mechanics viewpoint,” in Proceedings of the 1st JointConference of International Simulation Societies (CISS ’94), pp.661–666, Zurich, Switzerland, 1994.

[62] F. T. Brown, Engineering System Dynamics, CRC, Taylor andFrancis, London, UK, 2nd edition, 2007.

[63] A. Mukherjee, “The issue of invariants of motion for generalclass of symmetric systems through bond graph and umbra-Lagrangian,” in Proceedings of the International Conference onBond Graph Modeling and Simulation (ICBGM ’01), p. 295,Phoenix, Ariz, USA, 2001.

[64] V. Rastogi, Extension of Lagrangian-Hamiltonian mechanics,study of symmetries and invariants, Ph.D. thesis, IndianInstitute of Technology, Kharagpur, India, 2005.

[65] A. Mukherjee, V. Rastogi, and A. Dasgupta, “A methodologyfor finding invariants of motion for asymmetric systems withgauge-transformed umbra Lagrangian generated by bondgraphs,” Simulation, vol. 82, no. 4, pp. 207–226, 2006.

[66] A. Mukherjee, V. Rastogi, and A. Dasgupta, “A study ofa Bi-symmetric electro-mechanical system through Umbra-Lagrangian generated by bondgraphs, and Noether’s theo-rem,” Simulation, vol. 83, no. 9, pp. 611–630, 2007.

[67] A. Mukherjee, V. Rastogi, and A. Dasgupta, “Extension ofLagrangian-Hamiltonian mechanics for continuous systems-investigation of dynamics of a one-dimensional internallydamped rotor driven through a dissipative coupling,” Nonlin-ear Dynamics, vol. 58, no. 1-2, pp. 107–127, 2009.

[68] V. Rastogi and C. Kumar, “Investigation of dynamics of asym-metric rotor through Lagrangian formalism,” in Proceedings ofthe IUTAM Symposium on Emerging Trends in Rotor Dynamics(IUROTOR ’09), IIT Delhi, Delhi, India, March 2009.

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttp://www.hindawi.com Volume 2010

RoboticsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation http://www.hindawi.com

Journal ofEngineeringVolume 2014

Submit your manuscripts athttp://www.hindawi.com

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014

SensorsJournal of


Modelling & Simulation in EngineeringHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks


Review Article ABriefReviewonDynamicsofaCrackedRotor · 2.5. Analysis through Nonlinear Dynamics of...

Documents

Transcript of Review Article ABriefReviewonDynamicsofaCrackedRotor · 2.5. Analysis through Nonlinear Dynamics of...